Let $G=(V,E)$ be a locally finite graph, whose measure $μ(x)$ have positive lower bound, and $Δ$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely $Δu+hu=f(x,u)$, $x\in V$. In particular, we prove that if $h$ and $f$ satisfy certain assumptions, then the above mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation $Δu+hu=f(x,u)+εg$. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.
In this paper, we introduce the concept of local Leray solutions starting from a locally square-integrable initial data to the fractional Navier-Stokes equations with $s\in [3/4,1)$. Furthermore, we prove its local in time existence when $s\in (3/4, 1)$. In particular, if the locally square-integrable initial data vanishs at infinity, we show that the fractional Navier-Stokes equations admit a global-in-time local Leray solution when $s\in [5/6, 1)$. For such local Leray solutions starting from locally square-integrable initial data vanishing at infinity, the singularity only occurs in $B_R(0)…
We prove global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation we consider generalizes two-sphere-valued completely integrable systems modelling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an energy which is not necessarily preserved in time but does not blow up in finite time.
The century old extremal problem, solved by Carathéodory and Fejér, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the coefficient sequence (ck) will be supported in [-n; n] and normalized by c0 =1. Reformulating, nonnegativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c : Z --> C, supported in [-n; n]. Boas and Katz, Arestov, Berdysheva and Beren…
This paper is concerned with a fourth order nonlinear dispersive partial differential equation for closed curve flow on a Kähler manifold. The main results is that the initial value problem has a solution locally in time if the Kähler manifold is a compact locally hermitian symmetric space. The proof is based on the geometric energy method combined with a nice gauge transformation to eliminate the loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrödinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to en…
In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than $\frac {n-α}2$ for some $α\in [2, n)$, the set of positive smooth solutions to the equation $$ P_αu = u^\frac {n+α}{n-α} $$ is compact in the $C^\infty$ topology. There…
The current paper is concerned with positive stationary solutions and spatial spreading speeds of KPP type evolution equations with random or nonlocal or discrete dispersal in locally spatially inhomogeneous media. It is shown that such an equation has a unique globally stable positive stationary solution and has a spreading speed in every direction. Moreover, it is shown that the localized spatial inhomogeneity of the medium neither slows down nor speeds up the spatial spreading in all the directions.
This paper is devoted to the study of spatial propagation dynamics of species in locally spatially inhomogeneous patchy environments or media. For a lattice differential equation with monostable nonlinearity in a discrete homogeneous media, it is well-known that there exists a minimal wave speed such that a traveling front exists if and only if the wave speed is not slower than this minimal wave speed. We shall show that strongly localized spatial inhomogeneous patchy environments may prevent the existence of transition fronts (generalized traveling fronts). Transition fronts may exist in weak…
In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$α$ model and $p$-Laplace equation with non-monotone perturbations.
The lifespan of positive solutions of heat equation with power-logarithmic nonlinearity on locally finite graph
School of Mathematics, Renmin University of China（中国人民大学数学学院）；Fakultät für Mathematik, Universität Bielefeld（比勒费尔德大学数学系）；School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University（华中师范大学数学与统计学院及湖北省数学科学重点实验室）；Key Laboratory of Nonlinear Analysis & Applications (Ministry of Education), Central China Normal University（华中师范大学非线性分析与应用教育部重点实验室）
As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant scalar curvature metric. We provide different proofs using extension properties introduced by Chang and González (201…
This paper is devoted to the existence and non-existence of positive solutions for a $(p, q)$-Laplacian system with indefinite nonlinearity depending on two parameters $(λ,μ)$. By using the sub-supersolution method together with adaptations of ideas found in \cite{AAS,CRMZ}, we extend some previous result.
We study the existence of cylindrically symmetric electro-magneto-static solitary waves for a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations in presence of a positive mass and of a nonnegative nonlinear potential. Nonexistence results are provided as well.
In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison techniques, we show the existence, uniqueness, non-existence and regularity of the solutions. We also obtain a bifurcation-type result.
In this work we deal with the class of nonlinear (p,q)-Laplacian system. Non-existence results of positive weak solutions for this system are established.
In this paper I prove the existence of a positive stationary solution for a generic quasilinear model of structured population. The existence is proved using Schauder's fixed point theorem. The theorem is applied to a hierarchically size-structured population model.
We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$ condition, that is, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } …
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Other extensions related to bounded positive solutions to a schrodinger equation are also discussed.
Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDE for the film thickness and salt concentration subject to non-conservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and the results are then validated against PDE simulations. We also numerically capture other…
In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting properties of Schrodinger equation are derived.
In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.
Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show that, when the dimension is greater than or equal to 6, the Weyl tensor has to vanish at a point where solutions to the Yamabe equation blow up.
The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $γ\in [0,1)$ and satisfy $K(x,y) \leq C (x^γ + y^γ)$. More precisely, for any $ρ\in (γ,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}ρ)}$ as $x \to \infty$.
We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite $n\times n$ matrices (any quotient for $n=3$, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Lebesgue component. This is consistent with the conjecture that the limit measures are absolutely continuous.
In this paper, we give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system.
Given a closed countably $1$-rectifiable set in $\mathbb R^2$ with locally finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke flow starting from the given set with the following regularity property. For almost all time, the flow locally consists of a finite number of embedded curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of either 0, 60 or 120 degrees.
In this paper, we establish the results on the existence, nonexistence and multiplicity of positive solutions to singular boundary value problems involving $φ$-Laplacian. Our approach is based on the fixed point index theory. The interesting point is that a result for the existence of three positive solutions is given.
Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.
